Maple Lab for Calculus II
Lab B
Review: Limit, Derivative, Integral, Integration by Substitution
Douglas Meade, Ronda Sanders, and Xian Wu
Department of Mathematics
Overview
The goal of this week’s lab is to review basics covered in Calculus I such as limits, derivatives, integrals,
and integration by substitution using Maple.
Maple Essentials
• The Integration by Substitution maplet is available from the course website:
http://people.math.sc.edu/calclab/142L-S16/labs → Integration by Substitution
This maplet is designed to help you practice the steps involved in integration by substitution. Click on
Indefinite Integral or Definite Integral to have the maplet generate a problem for you to practice.
For a definite integral, you will be asked to calculate the associated indefinite integral first. You can then
choose to change the limits of integration or substitute to return to the original variable. You can click
Hint or Show if you need help or want to skip a step.
• Maple commands reviewed in this lab:
Command
:=x->
limit
Description
define a function of x. eg: f:=x->sqrt(x);
g:=t->2*sin(t);
compute limits or sided limits:
limit(f(x),x =a ); finds the limit, lim f (x).
x→a
limit(f(x),x =a,right); finds the one-sided limit from right, lim f (x).
x→a+
limit(f(x),x =a,left); finds the one-sided limit from left, lim− f (x).
x→a
diff
int
plot
compute derivatives of an expression:
diff(f(x), x ); finds the derivative of f (x) with respect to x.
diff(f(x), x$n ); finds the nth derivative of f (x).
compute definite and indefinite integrals:
R
int(f(x),x ); evaluates the indefinite integral, f (x) dx;
Rb
int(f(x),x =a..b ); evaluates the definite integral, a f (x) dx.
plot one or more functions on a specified window
plot(f(x), x =a..b ); plots the graph of y = f (x) for a < x < b;
plot([f(x),g(x) ], x =a..b ); graphs two functions in a single plot
Note that Maple’s int command does not include any constants of integration. Whenever you
evaluate an indefinite integral, do not forget to include a constant of integration (+C) in your
answer.
√
• Basic Functions and Expressions: Maple uses sqrt(x) for x, abs(x) for |x|, exp(x) for ex ,
and x^n for xn . Other basic functions can be typed in as what they are but you must carefully
use ( ) to group together and match up expressions as needed. You must also type in * when
multiplication is presented. For example, you need to type in sin(x) instead of sinx for sin x
2x
and 2*x/(x+y) instead of 2x/x+y for x+y
. Finally, do not forget to type in ; at the end of a line
that is to be executed.
• Palettes: The Expression and Calculus palettes can be used instead of typing the full Maple
commands for many functions and operations. The Common Symbols palette contains other
symbols, including π (Pi) and ∞ (infinity).
Related Course Material/Preparation
Review the basic terminology and properties of limits, derivatives, integrals, and integration by substition.
Spring 2016
Created 1/11/2016
Maple Lab for Calculus II
Lab B
Activities
Many of the problems below can be solved in Maple using different interface methods: typing in a
command directly, right-clicking over the expression, or using palettes. Try and/or combine different
methods when possible. You should make it a habit to define a function or assign an expression to a
name first. You should always inspect the output to verify that the function or expression was entered
correctly. You can then apply operations by right-clicking or referring to the name or the equation
label (if you use commands or palettes).
1. Launch Integration by Substitution maplet from the course web page and do a few practice problems.
2. Use Maple to compute the derivative of the following functions:
1 − ln(x)
arctan(x)
(iii) h(x) =
(i) f (x) = x3 ln(x)
(ii) g(x) = √
2
1 + ln(x)
x +1
Notice that if you use a right-click to obtain the derivative the resulting derivative will be defined as
a function. If you use the diff command (or the Calculus palette) the resulting derivative will be an
expression.
3. Use Maple to compute the first, second, third, and 100th derivatives of f (x) = x sin(2x).
4. Use Maple to evaluate the following indefinite and definite integrals:
Z 3
Z
Z π/2 1
x+1
−x2
dx
dx
(ii)
−
cos
x
(i)
xe
dx
(iii)
(x − 2)2
2
−1
0
5. Use Maple to evaluate the following limits:
π x2 arccos(x)
2n
(i) lim
(ii)
lim
(1
+
3/n)
(iii)
lim
tan
t
n→∞
x→0 (sin(2x))2
2
t→1+
6. Let
1
f (x) = x4 − x3 − 36x2 + 108x
2
In a Maple worksheet:
(a) Define the function using proper Maple notation.
(b) Use right-clicking to find the first and second derivatives of f (x). Assign the derivatives to
names using := with appropriate label references or by using right-clicking.
Hint: Reasonable names for these quantities would be df and ddf.
0
00
(c) Find f (1), f (1), and f (1). Explain what each of these values tells you about the function
f (x) when x = 1.
(d) Plot the graph of y = f 0 (x) on the interval −10 < x < 10 and estimate the intervals on which
y = f (x) is increasing. Note: You should adjust the window to get a good picture of the graph.
(e) Plot the graph of y = f 00 (x) on the interval −10 < x < 10 and estimate the intervals on which
y = f (x) is concave up.
(f) Create a single plot containing the graphs of y = f (x), y = f 0 (x), and y = f 00 (x) for
−10 < x < 10. Did you estimate correctly in (d) and (e)?
(g) Don’t forget to give your graph a title and legend and make sure the curves can be identified
on a black/white copy. (Review Calc I Lab C if necessary.) You can achieve this using
commands or by right-clicking on your graph to see the context menu. Remember if you use
right-clicking that your changes will be lost if you re-execute.
(h) For some Maple assignments/projects it will be useful to copy your final plot to a text document such as the Word. See if you are able to do this now.
Assignment
With the help of Maple, work out the problems assigned by your lab instructor. Clearly identify your
answers on your Maple worksheet. Make sure you answer each question completely. Your assignment
is due at the beginning of next week’s lab.
Spring 2016
Created 1/11/2016